IUMJ

Title: Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

Authors: Thomas Duyckaerts and Frank Merle

Issue: Volume 58 (2009), Issue 4, 1971-2002

Abstract:

We consider the energy-critical semilinear focusing wave equation in dimension $N = 3, 4, 5$. An explicit solution $W$ of this equation is known. By the work of C. Kenig and F. Merle, any solution of initial condition $(u_0, u_1)$ such that $E(u_0, u_1) < E(W,0)$ and $\|\nabla u_0\|_{L^2} < \|\nabla W\|_{L^{2}}$ is defined globally and has finite $L^{(2(N+1))/(N-2)}_{t,x}$-norm, which implies that it scatters. In this note, we show that the supremum of the $L^{(2(N+1))/(N-2)}_{t,x}$-norm taken on all scattering solutions at a certain level of energy below $E(W,0)$ blows-up logarithmically as this level approaches the critical value $E(W,0)$. We also give a similar result in the case of the radial energy-critical focusing semilinear Schr\"odinger equation. The proofs rely on the compactness argument of C. Kenig and F. Merle, on a classification result, due to the authors, at the energy level $E(W,0)$, and on the analysis of the linearized equation around $W$.